Summations in Summer

Number Theory Level 3

There is a unique function $f : \mathbb N \rightarrow \mathbb R$ such that $f(1) > 0$ and such that

$\large\ \displaystyle \sum_{d|n} f(d)f\left( \frac { n }{ d } \right) = 1$

for all $n \ge 1$ .

What is $\large\ f\left( { 2018 }^{ 2019 } \right)$ ?

\frac { { \left( 4038! \right) }^{ 2 } }{ { \left( 2019! \right) }^{ 4 }\cdot { 2 }^{ 8076 } }

\frac{{\left( 4038! \right) }^{ 4 } }{ { \left( 2019! \right) }^{ 2 }\cdot { 2 }^{ 8076 } }

\frac{{\left( 4038! \right) }^{ 2 } }{ { \left( 2019! \right) }^{ 2 }\cdot { 2 }^{ 8076 } }

\frac{{\left( 4038! \right) }^{ 4 } }{ { \left( 2019! \right) }^{ 4 }\cdot { 2 }^{ 8076 } }

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